The value of $a$ such that $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$ decreases for all real values of $x$ is given by:

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speed $\omega_1$ and $\omega_2$ are brought into contact face to face with their axes of rotation coincident. What is the loss in kinetic energy of the system in the process?

$A$ wheel is rotating at $900 \ r.p.m.$ about its axis. When the power is cut-off,it comes to rest in $1 \ minute$. The angular retardation in $rad/s^2$ is

Which enzyme stimulates the germination of barley seeds?

Let $A = \{x \in R, x \neq 0, -4 \leq x \leq 4\}$ and $f: A \rightarrow R$ be defined by $f(x) = \frac{|x|}{x}$ for $x \in A$. Then,the range of $f$ is

If the distance of the point $P(1, -2, 1)$ from the plane $x + 2y - 2z = \alpha$,where $\alpha > 0$,is $5$,then the foot of the perpendicular from $P$ to the plane is: